Fyodor A. Sukochev

Nurulla A. Azamov

Comptes Rendus Mathématique

1/dicl-acesriodiques)sproche-IC. R. Acad. Sci. Paris, Ser. I 340 (2005) 107–112http://france.elsevier.com/direct/CRASSMathematical AnalysisA Lidskii type formula for Dixmier tracesNurulla A. Azamov, Fyodor A. SukochevSchool of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042, SA, AustraliaReceived 15 September 2004; accepted after revision 30 November 2004Presented by Alain ConnesAbstractWe present a new formula to compute Dixmier tracesτω(T ) of pseudodifferential operators (respectively, almost periopseudodifferential operators) of order−n onn-dimensional compact Riemannian manifolds (respectively,Rn). Under a naturacondition on the operatorT , we show thatτω(T ) = ω-limt→∞ 1log(1+t )∫λ/∈ 1t G λdµT (λ), whereG is any bounded neighborhood of 0∈ C andµT is the Brown spectral measure ofT . If T is measurable, then theω-limit may be replaced withthe true (ordinary) limit. Our approach works equally well in both type I and II settings.To cite this article: N.A. Azamov,F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméUne nouvelle formule pour calculer les traces de Dixmier. Nous présentons une nouvelle formule pour calculer les trde Dixmierτω(T ) des opérateurs pseudodifférentiels (respectivement, des opérateurs pseudodifférentiels presque péd’ordre−n sur des variétés compactes de dimensionn (respectivement,Rn). LorsqueT satisfait une condition naturelle, noumontrons queτω(T ) = ω-limt→∞ 1log(1+t )∫λ/∈ 1tGλdµT (λ), où G est un voisinage borné de 0 dansC et µT est la mesurespectrale de Brown deT . Si T est mesurable, on peut remplacer la limite faible par la limite au sens usuel. Notre aps’applique aux types I et II.Pour citer cet article : N.A. Azamov, F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégéeD’après le théorème de Lidskii (sous la forme générale semi-finie de [1]), siN est un facteur de Von Neumann semi-fini muni d’une trace fidèle, normale, semi-finie, la traceτ (T ) d’un opérateurT ∈ L1(N , τ ) estτ (T ) = ∫σ(T )\{0} λdµT (λ), où µT est la mesure spectrale de Brown deT . LorsqueN est un facteur de type(respectivement, lorsqueT est un opérateur normal)µT est la mesure de comptage sur l’ensemble {λn(T )}∞n=1des valeurs propres deT (respectivement, la mesureτ -spectrale deT , donnée parµT (B) = τ (χB(T )) surE-mail addresses:azam0001@infoeng.flinders.edu.au (N.A. Azamov), sukochev@infoeng.flinders.edu.au (F.A. Sukochev).1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2004.12.005108 N.A. Azamov, F.A. Sukochev / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 107–112nsitvaleursffet,s cetteuect-eratorsliodicnles boréliensB ⊂ C). Nous donnons ici une formule analogue pour les traces de Dixmier. Nous définissoL(1,∞) = {T ∈ N : supt>0 1log(1+t )∫ t0 µs(T )ds < ∞}, oùµt (T ) est lat-ième valeur singulière deT [10]. Lorsqueω est un état surL∞(0,∞), s’annulant sur les fonctions à support compact et satisfaisantω(Mf ) = ω(f )pour toutef ∈ L∞(0,∞), avecM(f )(t) = 1log(1+t )∫ t0 f (s)dss, t ∈ (0,∞), la trace de Dixmier [5,8]τω(T ) estτω(T ) = ω-limt→∞ 1log(1+t )∫ t0 µs(T )ds pourT ∈ L(1,∞), positif et s’étend par linéarité dans le cas général.Dans le cas classique, siT est un opérateur à trace, compact, auto-adj int, le théorème de Lidskii se dédudirectement du théorème spectral pourles opérateurs positifs et de la convergence absolue de la série despropres deT . Si T = T ∗ est dansL(1,∞), cette série peut diverger, ce qui est un obstacle non trivial. En enous montrerons comme cas particulier de notre résultat principal que siT = T ∗ ∈ L(1,∞) vérifie |λn(T )| Cnpour unC > 0 et toutn  1, avec |λ1(T )|  |λ2(T )|  · · · si τω est une trace de Dixmier, alorsτω(T ) =ω-limN→∞ 1log(1+N)∑Nn=1 λn(T ). De plus, siT est mesurable au sens de Connes [5], on peut remplacer danformule la limite faible par la limite au sens usuel.1. IntroductionThe well-known Lidskii theorem (in its general semifinite form given in [1]) asserts that ifN is a semifinite vonNeumann factor with a faithful normal semifinite traceτ, then the traceτ (T ) of an arbitrary operatorT ∈ L1(N , τ )is given byτ (T ) = ∫σ(T )\{0} λdµT (λ), whereµT is the Brown’s measure ofT . In the case whenN is a type I fac-tor, the measureµT is the counting measure on the set of all eigenvalues ofT . In this paper, we present an analogof such a formula for Dixmier traces. Letω be a state onL∞(0,∞) which vanishes on functions with compasupport and such thatω(Mf ) = ω(f ) for everyf ∈ L∞(0,∞), whereM(f )(t) = 1log(1+t )∫ t0 f (s)dss. It will beconvenient to writeω-limt→∞ f (t) instead ofω(f ), f ∈ L∞(0,∞). The Dixmier trace [5,6]τω(T ) defined onthe idealL(1,∞) := {T ∈ N : supt>0 1log(1+t )∫ t0 µs(T )ds < ∞} is given byτω(T ) = ω-limt→∞ 1log(1+t )∫ t0 µs(T )dsif T ∈ L(1,∞) is positive and by linearity otherwise. Here,µt (T ) = inf{‖T E‖: E is a projection inN withτ (1− E)  t} is thet th generalizeds-number of the operatorT .In the case of a standard (normal) trace, the assertion of the Lidskii theorem for self-adjoint operators is immediate due to the absolute convergence of the series∑n1 λn(T ) of anyT = T ∗ from the trace class. This is not thcase any longer for Dixmier (non-normal) traces, since the latter series diverges for anyT = T ∗ ∈ L(1,∞) whichdoes not belong to the trace class.The distribution function ofT ∈ N is defined byλt (T ) := τ (χ(t,∞)(|T |)), t > 0. We haveµs(T ) =inf{t  0: λt (T )  s} and for anys, t > 0, s  λt (T ) if and only if µs(T )  t . Furthermore,∫ λt (T )0 µs(T )ds =τ (|T |χ(t,∞)(|T |)), ∀t > 0. These facts may be found in [2,10]. We writeS ≺≺ T iff∫ t0 µs(S)ds ∫ t0 µs(T )ds,∀t > 0.Our main result is given in Theorem 2.11 below. The Lidskii type formula given there holds for all opeT ∈ L(1,∞) satisfyingµt(T )  C/t for someC > 0 and allt > 0. The set of such operators form an ideal inNdenoted byL1,w. The idealL1,w usually arises in geometric applications. In particular, ifN is the algebra of albounded operators onL2(M) whereM is a compact Riemanniann-manifold (respectively, ifN is theII ∞-factorL∞(Rn)  Rndiscr [4,12]), the idealL1,w contains all pseudodifferential operators (respectively, all almost perpseudodifferential operators) of order−n.An operatorT fromL(1,∞) is said to be measurable ifτω(T ) does not depend on the stateω [5]. For an arbitrarysubsetA ⊆ N , we denote byAm the set of measurable elements fromA.Our formula takes an especially simple form for the case of measurable operatorsT . In this case,τω(T ) coin-cides with the true limit limt→∞ 1log(1+t )∫λ/∈ 1tG λdµT (λ) for an arbitraryω.Our results depend crucially on the recent characterization of positive measurable operators fromL(1,∞) asthose for which the limit limt→∞ 1log(1+t )∫ t0 µs(T )ds exists [11, Theorem 6.6], and the spectral characterizatioof sums of commutators in type II factors [8,9].N.A. Azamov, F.A. Sukochev / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 107–112 109or-yi-on-2. Lidskii formulae for Dixmier tracesLemma 2.1. If T  0 in L1,w then there existsS ∈ L1,wm such thatT  S, supp(S)  supp(T ) andST = T S.Proof. If N is a type II factor, then the assertion follows from [7, Proposition 3.2]. The type I case is straightfward. The proof of the following lemma is straightforward and is therefore omitted.Lemma 2.2. If T ∈L(1,∞)m thenT ∗, Re(T ), Im(T ) ∈L(1,∞)m . The same assertion also holds forL1,w.Remark 1. The positive and negative parts of a measurable self-adjoint operatorT ∈ L(1,∞) are not necessarilmeasurable.Lemma 2.3 (see [2]). For T ∈ L(1,∞), we haveλ1/t (T )  Ct logt, for someC > 0, and all sufficiently larget .For brevity, we writeft (T ) =∫ t0 µs(T )ds andgt (T ) =∫ λ1/t (T )0 µs(T )ds, t > 0. The results given in Propostion 2.4 and Lemma 2.5 below are similar to those obtained [2, Proposition 2.4] under different assumptionsω andT .Proposition 2.4. If T  0 in L1,w, then for everyC > 0τω(T ) = ω- limt→∞1log(1+ t) ft (T ) = ω- limt→∞1log(1+ t)Ct logt∫0µs(T )dsand if one of theω-limits is a true limit then so is the other.Proof. For T  0 in L1,w, we have 1log(1+t ) |∫ Ct logt0 µs(T )ds − ft (T )|  Mlog(1+t ) (log(Ct logt) − logt) → 0 ast → ∞ for someM > 0. The second assertion is proved in [2, Proposition 2.4].Lemma 2.5. If T  0 in L1,w thenτω(T ) = ω- limt→∞1log(1+ t) gt (T ). (1)If T is measurable then theω-limit can be replaced with the true limit.Proof. It follows from Lemma 2.3 and Proposition 2.4 thatω- limt→∞1log(1+ t)gt (T )  ω- limt→∞1log(1+ t)Ct logt∫0µs(T )ds = τω(T ). (2)Further, sinces > λ1/t impliesµs(T )  1/t, we haveft (T )  gt (T ) + 1t (t − λ1/t (T ))  gt (T ) + 1. Hence,τω(T )  ω- limt→∞1log(1+ t) gt (T ). (3)Combining (2) and (3), we arrive at (1). The second assertion follows from Proposition 2.4 and [11, Theorem 6.6]. Lemma 2.6. If A,B,C  0 in L1,w andC = A + B then110 N.A. Azamov, F.A. Sukochev / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 107–112proof5s(i) limt→∞|ft (A) + ft (B) − ft (C)|log(1+ t) = 0, (ii) limt→∞|gt (A) + gt (B) − gt (C)|log(1+ t) = 0. (4)Proof. Let A′  0 andB ′  0 be operators fromL(1,∞) such thatµt(A′) = µt(A), µt (B ′) = µt(B) ∀t > 0 andA′B ′ = 0. Let C′ = A′ + B ′. According to [3, Lemma 2.3] we haveC′ ≺≺ C. Combining this fact with theobservation thatµ(C) ≺≺ µ(A) + µ(B) [10, Proposition 2.4], we see that it is sufficient to prove thatlimt→∞1log(1+ t)∣∣ft (A′) + ft (B ′) − ft (C′)∣∣ = 0. (5)SinceA′ andB ′ are orthogonal, we havegt (A′) + gt (B ′) − gt (C′) = 0. (6)Combining (6) with Lemma 2.3 and the argument in the proof of Proposition 2.4 we arrive at (4(i)). Theof (4(ii)) is similar. Lemma 2.7. Let T ∈ L1,w be normal andT = T1 − T2 + iT3 − iT4, where T1, . . . , T4  0. Then τω(T ) =ω-limt→∞ 1log(1+t ) (gt (T1) − gt (T2) + igt (T3) − igt (T4)). If T is measurable then theω-limit can be replaced withthe true limit.Proof. The first assertion follows from Lemma 2.5 and the linearity of Dixmier traces. LetT be measurableand self-adjoint. LetS  0 in L(1,∞) be a measurable operator commuting withT− such thatS − T−  0 andsupp(S)  supp(T−) (see Lemma 2.1). We haveτω(T ) = τω(S + T ) − τω(S) = ω-limt→∞ 1log(1+t )ft (S + T ) −ω-limt→∞ 1log(1+t )ft (S). Since 0 T ,T + S are measurable, we may combine [11, Theorem 6.6] with Lemma 2.to obtainτω(T ) = limt→∞ 1log(1+t )gt (S+T )− limt→∞ 1log(1+t )gt (S). SinceS+T = S−T−+T+ and the operatorS − T− andT+ are disjoint, we havegt (S + T ) = gt (S − T−) + gt (T+). Eq. (4(ii)) of Lemma 2.6 withA = T−,B = S −T− andC = S now yields limt→∞ 1log(1+t ) (gt (T+)−gt (T−)−gt (S +T )+gt (S)) = 0. Taking theω-limit,we conclude from [11, Theorem 6.6] thatτω(T ) = ω- limt→∞gt (T+) − gt (T−)log(1+ t) = ω- limt→∞gt (S + T ) − gt (S)log(1+ t)= limt→∞gt (S + T ) − gt (S)log(1+ t) = limt→∞gt (T+) − gt (T−)log(1+ t) . (7)If T is normal, the assertion now follows from Lemma 2.2.Lemma 2.8. If T ∈ L1,w is normal anda > 0 thenτω(T ) = ω-limt→∞ 1log(1+t )∫λ/∈Qt λdµT (λ), whereQt = {x +iy ∈ C: |tx|  a, |ty|  a} ∀t > 0. If T is measurable, then theω-limit can be replaced with the true limit.Proof. We may takea = 1, by dilation invariance ofω [5, p. 305]. LetT = T1−T2+ iT3− iT4, whereT1, . . . , T4 0. ForT  0 in L1,w, we havegt (T ) =∫ ∞1/t λdµT (λ) (see, e.g. [2]). LetĀ be the complement ofA ⊆ C, Rt :={λ ∈ C : |Re(λ)|  1/t} andIt := {λ ∈ C: |Im(λ)|  1/t}. For any Borel setB ⊆ R, we have∫B λdµRe(T )(λ) =∫{λ: Re(λ)∈B} Re(λ)dµT (λ),∫BλdµIm(T )(λ) =∫{λ: Im(λ)∈B} Im(λ)dµT (λ) and so∫Q̄tλdµT (λ) =∫Q̄tRe(λ)dµT (λ) + i∫Q̄tIm(λ)dµT (λ)=∫R̄Re(λ)dµT (λ) +∫Q̄ ∩RRe(λ)dµT (λ) + i∫ĪIm(λ)dµT (λ) + i∫Q̄ ∩IIm(λ)dµT (λ)t t t t t tN.A. Azamov, F.A. Sukochev / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 107–112 111usly.r-ees.10.=∫{|ξ |>1/t}ξ dµRe(T )(ξ) +∫Q̄t∩RtRe(λ)dµT (λ) + i∫{|ξ |>1/t}ξ dµIm(T )(ξ) + i∫Q̄t∩ItIm(λ)dµT (λ).The sum of the first and the third terms in the expression above givesτω(T ) after taking theω-limitwith respect tot → ∞ (see Lemma 2.7). We shall now show that limt→∞ 1log(1+t )∫Q̄t∩Rt Re(λ)dµT (λ) = 0and limt→∞ 1log(1+t )∫Q̄t∩Rt Im(λ)dµT (λ) = 0. We prove the first equality, the second is proved analogoIn fact, it suffices to prove that limt→∞ 1log(1+t )∫{Im(λ)>1/t}∩Rt Re(λ)dµT (λ) = 0. We have∣∣∣∣∫{Im(λ)>1/t}∩RtRe(λ)dµT (λ)∣∣∣∣  1t∫{Im(λ)>1/t}∩RtdµT (λ) 1t∫{Im(λ)>1/t}dµT (λ) = 1t∞∫1/tdµIm(T )(ξ)= 1tτ(χ(1/t,∞)(T3)) = 1tλ1/t (T3)  C.The last inequality follows from the equivalence ofµCt(T3)  1/t andλ1/t (T3)  Ct. For the case of measuableT , the proof is the same.Lemma 2.9. Let T be a normal operator fromL1,w and letG be a bounded Borel neighborhood of0 ∈ C. Thenτω(T ) = ω-limt→∞ 1log(1+t )∫λ/∈ 1t G λdµT (λ). If T is measurable, then theω-limit can be replaced with the trulimit.Proof. For an arbitrary bounded neighborhoodG of 0 ∈ C there exist squaresQa andQb such thatQa ⊆ G ⊆ Qb.Hence, Lemma 2.8 implies that it is sufficient to prove thatlimt→∞1log(1+ t)∫Qt/b\Qt/a|λ|dµT (λ) = 0. (8)The setQt/b \ Qt/a consists of four trapeziums and it is suffices to prove the above limit for one of them,Dt :={z ∈ Qt/b \ Qt/a: Re(tz) ∈ [a, b]}, for example. We have12∫Dt|λ|dµT (λ) ∫DtRe(λ)dµT (λ) ∫t Re(λ)∈[a,b]Re(λ)dµT (λ) =b/t∫a/tλdµRe(T )(λ) =λa/t∫0µs ds −λb/t∫0µs ds.By Lemma 2.5, we can replace upper limitsλa/t and λb/t by t/a and t/b respectively. Then∫ t/a0 µs ds −∫ t/b0 µs ds ∫ t/at/b C/s ds  C logba. The following lemma follows from [8].Lemma 2.10. If S ∈ L1,w then there exists a normal operatorT ∈ L(1,∞) such that the Brown spectral measurof S andT coincide andτω(S) = τω(T ).The proof of the following theorem (which is the main result of this note) follows from Lemmas 2.9 and 2Theorem 2.11. If S ∈ L1,w andG is a bounded Borel neighborhood of0 ∈ C, thenτω(S) = ω-limt→∞ 1log(1+t ) ×∫λ/∈ 1t G λdµS(λ). If S is measurable, then theω-limit can be replaced with the true limit.We specialize the result above to the case whenN is an infinite-dimensional factor of typeI∞.112 N.A. Azamov, F.A. Sukochev / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 107–112tient totrf or-f of,1997)9–roc.ittedCorollary 2.12. Let T be a compact operator on an infinite-dimensional Hilbert spaceH such thatµn(T ) C/n, n  1, for someC > 0. If λ1, λ2, . . . is the list of eigenvalues ofT counting the multiplicities such tha|λ1|  |λ2|  · · · , thenTrω(T ) = ω- limt→∞1log(1+ t)∑λ∈σ(T ), λ/∈ 1t GλµT (λ) = ω- limN→∞1logNN∑i=1λi,whereµT (λ) is the algebraic multiplicity of the eigenvalueλ. If T is measurable then theω-limit can be replacedwith the true limit.Proof. The first equality is an immediate consequence of Theorem 2.11. By Lemma 2.10, it is sufficprove the second equality for a normal operatorT . Let G := {z ∈ C: |z| < 1}. It is enough to show tha∑k∈AN∪BN |λk| < const, whereAN = {k ∈ N: k  N, |λk |  1/N} andBN = {k ∈ N: k > N, |λk| > 1/N}. Wehave,∑k∈AN |λk|  1. That∑k∈BN |λk| is bounded follows from the condition|λk| < C/k, k ∈ N, for someC > 0 and estimate (8). The following corollary follows from the combination of Corollary 2.12 and [5, Proposition IV.2.5].Corollary 2.13 [9, Proposition 1]. If M is a compact Riemanniann-manifold andT is a pseudodifferential operatoof order−n onM, thenTrω(T ) = limN→∞ 1logN∑Nk=1 λk .Let N = L∞(Rn)  Rndiscr and letT  be (the image of) an almost periodic pseudodifferential operator oder−n (see, for example, [12]). ThenT  ∈L1,w.Corollary 2.14. τω(T ) = ω-limt→∞ 1log(1+t )∫|λ|>1/t λdµT (λ).AcknowledgementsThe authors thank Nigel Kalton for explaining results from [8], in particular for indicating the prooLemma 2.10. We also thank Thierry Fack for useful discussions.References[1] L.G. Brown, Lidskii’s theorem in the type II case, in: Proc. U.S.–Japan Seminar, Kyoto 1983, PitnamRes. Notes Math. Ser., vol. 1231986, pp. 1–35.[2] A.L. Carey, J. Phillips, F.A. Sukochev, Spectral flow and Dixmier traces, Adv. Math. 173 (2003) 68–113.[3] V.I. Chilin, P.G. Dodds, F.A. Sukochev, The Kadec–Klee property in symmetric spaces of measureable operators, Israel J. Math. 97 (203–219.[4] L.A. Coburn, R.D. Moyer, I.M. Singer,C∗-algebras of almost periodic pseudo-differential operators, Acta Math. 130 (3–4) (1973) 27307.[5] A. Connes, Non-Commutative Geometry, Academic Press, San Diego, 1994.[6] J. Dixmier, Existence de traces non-normales, C. R. Acad. Sci. Paris A-B 262 (1966) A1107–A1108.[7] P.G. Dodds, F.A. Sukochev, G. Schluchtermann, Weak compactnesscriteria in symmetric spaces of measurable operators, Math. PCambridge Philos. Soc. 131 (2001) 363–384.[8] K.J. Dykema, N.J. Kalton, Sums of commutators inideals and modules of type II factors, preprint.[9] T. Fack, Sums of commutators in non-commutativeBanach function spaces, J. Funct. Anal. 207 (2004) 358–398.[10] T. Fack, H. Kosaki, Generaliseds-numbers ofτ -measurable operators, Pacific J. Math. 123 (1986) 269–300.[11] S. Lord, A. Sedaev, F. Sukochev, Dixmier traces as singular symmetric functionals and applications to measurable operators, submfor publication.[12] M.A. Shubin, Pseudodifferential almost periodic operators and von Neumann algebras, T. Moscow Math. Soc. 1 (1979) 103–166.

A Lidskii type formula for Dixmier traces

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A Lidskii type formula for Dixmier traces

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